Abstract
Every real-world physical problem is inherently based on uncertainty. It is essential to model the uncertainty then solve, analyze and interpret the result one encounters in the world of vagueness. Generally, science and engineering problems are governed by differential equations. But the parameters, variables and initial conditions involved in the system contain uncertainty due to the lack of information in measurement, observations and experiment. However, It is necessary to develop a comprehensive approach for solving differential equations in an uncertain environment. The purpose of this work is to study and investigate the fuzzy solution of linear third-order fuzzy differential equations using the concept of strongly generalized Hukuhara differentiability (SGHD). To make our analysis possible, we apply the first and second differentiability up to the third-order fuzzy derivative of the fuzzy-valued function. Moreover, we develop an important result concerning the relationship between Laplace transform of fuzzy-valued function and third-order derivative. We construct an algorithm to determine a potential solution of linear third-order fuzzy initial-value problem using the Laplace transform technique. All these solutions are represented in terms of the Mittag-Leffler function involving a single series. Furthermore, we discuss the switching points of linear third-order differential equations and their corresponding solutions in fuzzy environments. To enhance the novelty of the proposed technique, some illustrative examples are presented as applications are analyzed to visualize and support theoretical results.
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References
Ahmadi MB, Kiani NA, Mikaeilvand N (2014) Laplace transform formula on fuzzy nt\(H\)-order derivative and its application in fuzzy ordinary differential equations. Soft Comput 18(12):2461–2469
Akram M, Ihsan T (2022) Solving Pythagorean fuzzy partial fractional diffusion model using the Laplace and Fourier transforms. Granul Comput. https://doi.org/10.1007/s41066-022-00349-8
Akram M, Saqib M, Bashir S et al (2022) An efficient numerical method for solving \(m\)-polar fuzzy initial value problems. Comp Appl Math 41:157
Akram M, Ihsan T, Allahviranloo T, Al-Shamiri MMA (2022) Analysis on determining the solution of fourt\(H\)-order fuzzy initial value problem with Laplace operator. Math Biosci Eng 19(12):11868–11902
Akram M, Muhammad G, Allahviranloo T, Ali G (2022) New analysis of fuzzy fractional Langevin differential equations in Caputo’s derivative sense. AIMS Math 7(10):18467–18496
Akram M, Muhammad G, Allahviranloo T, Ali G (2023) A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations. AIMS Math 8(1):228–263
Akram M, Muhammad G (2022) Analysis of incommensurate multi-order fuzzy fractional differential equations under strongly generalized fuzzy Caputos differentiability. Granul Comput
Allahviranloo T (2020) Fuzzy fractional differential operators and equations: fuzzy fractional differential equations, vol 397. Springer Nature, Berlin
Allahviranloo T, Ahmadi MB (2010) Fuzzy Laplace transforms. Soft Comput 14(3):235–243
Allahviranloo T, Pedrycz W (2020) Soft numerical computing in uncertain dynamic systems. Academic Press, New York. https://doi.org/10.1016/B978-0-12-822855-5.00001-X
Allahviranloo T, Soheil S (2022) Advances in fuzzy integral and differential equations. Springer, Berlin
Allahviranloo T, Kiani NA, Barkhordari M (2009) Toward the existence and uniqueness of solutions of second-order fuzzy differential equations. Inf Sci 179(8):1207–1215
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151(3):581–599
Bede B, Stefanini L (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141
Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern B Cybern 1:30–34
Chehlabi M (2018) Continuous solutions to a class of first-order fuzzy differential equations with discontinuous coefficients. Comput Appl 37(4):5058–5081
Chen M, Wu C, Xue X, Liu G (2008) On fuzzy boundary value problems. Inf Sci 178(7):1877–1892
Citil HG (2019) Investigation of a fuzzy problem by the fuzzy Laplace transform. Appl math nonlinear sci 4(2):407–416
Dubios D, Prade H (1982) Towards fuzzy differential calculus part 3: Differentiation. Fuzzy Sets Syst 8(3):225–233
ElJaoui E, Melliani S (2015) Chadli LS (2015) Solving second-order fuzzy differential equations by the fuzzy Laplace transform method. Adv Differ Equ 1:1–14
Friedman M, Ming M, Kandel A (1996) Fuzzy derivatives and fuzzy Cauchy problems using LP metric. In Fuzzy Logic Foundations and Industrial Applications 8:57–72
Goetschel R, Voxman W (1986) Elementary calculus. Fuzzy Sets Syst 18(1):31–43
Gumah G (2022) Numerical solutions of special fuzzy partial differential equations in a reproducing kernel Hilbert space. Comput Appl 41(2):1–17
Hoa NV (2015) The initial value problem for interval-valued second-order differential equations under generalized \(H\)-differentiability. Inf Sci 311:119–148
Hoa NV (2020) On the initial value problem for fuzzy differential equations of non-integer order \(\alpha \in (1, 2)\). Soft Comput 24(2):935–954
Hoa NV, Tri PV, Dao TT, Zelinka I (2015) Some global existence results and stability theorem for fuzzy functional differential equations. J Intell Fuzzy Syst 28(1):393–409
Hoa NV, Lupulescu V, O’Regan D (2017) Solving interval-valued fractional initial value problems under Caputo \(gH\)-fractional differentiability. Fuzzy Sets Syst 309:1–34
Ji LY, You CL (2021) Milstein method for solving fuzzy differential equation. Iran J Fuzzy Syst 18(3):129–141
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24(3):301–317
Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Sets Syst 35(3):389–396
Karimi F, Allahviranloo T, Pishbin SM, Abbasbandy S (2021) Solving Riccati fuzzy differential equations. New Math Nat Comput 17(01):29–43
Liu R, Fevckan M, Wang J, O’Regan D (2021) Ulam type stability for first-order linear and nonlinear impulsive fuzzy differential equations. Int J Comput Math 99(6):1–20
Mazandarani M, Xiu L (2021) A review on fuzzy differential equations. IEEE Access 9:62195–62211
Melliani S, Belhallaj Z, Elomari M, Chadli LS (2021) Approximate solution of intuitionistic fuzzy differential equations with the linear differential operator by the homotopy analysis method. Adv Fuzzy Syst. https://doi.org/10.1155/2021/5579669
Najariyan M, Qiu L (2021) Interval type-2 fuzzy differential equations and stability. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2021.3097810
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier Science, Amsterdam
Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91(2):552–558
Salahshour S, Allahviranloo T (2013) Applications of fuzzy Laplace transforms. Soft Comput 17(1):145–158
Salamat N, Mustahsan M, Missen MS (2019) Switching Point solution of second-order fuzzy differential equations using differential transformation method. Mathematics 7(3):231
Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330
Song S, Wu C (2000) Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations. Fuzzy Sets Syst 110(1):55–67
Xu J, Liao Z, Hu Z (2007) A class of linear differential dynamical systems with fuzzy initial condition. Fuzzy Sets Syst 158(21):2339–2358
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
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Akram, M., Muhammad, G., Allahviranloo, T. et al. Solution of initial-value problem for linear third-order fuzzy differential equations. Comp. Appl. Math. 41, 398 (2022). https://doi.org/10.1007/s40314-022-02111-x
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DOI: https://doi.org/10.1007/s40314-022-02111-x
Keywords
- Switching points
- Fuzzy initial-value problem
- Strongly generalized differentiability
- Fuzzy Laplace transform
- Mittag-Leffler function in single series